Shhin Fghiri , Shhin Akri , Mohmmd Behshd Shfii , Kh. Hosseinzdeh ,

1 Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

b Sharif Energy, Water and Environment Institute (SEWEI), Tehran, Iran

Keywords: Non-newtonian fluid Power-law model Non-uniform heat flux Analytical solution

ABSTRACT The present article aims to investigate the Graetz-Nusselt problem for blood as a non-Newtonian fluid obeying the power-law constitutive equation and flowing inside the axisymmetric tube subjected to non- uniform surface heat flux. After the flow field is determined by solving the continuity and the momentum equations, the energy equation is handled by employing the separation of variables method. The result- ing Eigen functions and Eigen values are numerically calculated using MATLAB built-in solver BVP4C. The analysis is first conducted for the situation of constant heat flux and subsequently generalized to apply to the case of sinusoidal variation of wall heat flux along the tube length, using Duhamel’s Theorem. Fur- thermore, an approximate analytic solution is determined, employing an integral approach to solve the boundary layer equations. With respect to the comparison, the results of approximate solution display acceptable congruence with those of exact solution with an average error of 7.4%. Interestingly, with de- creasing the power-law index, the discrepancy between the two presented methods significantly reduces. Eventually, the influences of the controlling parameters such as surface heat flux and power-law index on the non-Newtonian fluid flow’s thermal characteristics and structure are elaborately discussed. It is found that switching from constant wall heat flux to non-uniform wall heat flux that sinusoidally varies along the tube length significantly improves the simulation’s accuracy due to the better characterization of the heat transport phenomenon in non-Newtonian fluid flow through the tube. In the presence of si- nusoidally varying wall heat flux with an amplitude of 200 W/m 2 and when the power-law index is 0.25, the maximum arterial wall temperature is found to be about 311.56 K.

Non-Newtonian fluids are described by the nonlinear relation- ship between shear stress and rate of deformation at a specified temperature and pressure [1] . The flows of non-Newtonian flu- ids play important role in many industrial applications and disci- plinary fields such as biomedicine, polymer and food processing, thermal oil recovery, and discharge of industrial wastes [2–8] . From a mechanical engineering viewpoint, the complicated rheological behavior of shear-thinning fluid like blood cannot be modeled by a very plain, one parameter, and linearized law of viscosity as pre- sented by Newton [9–13] . The properties of this type of fluid can only be characterized by higher-order constitutive equations, such as the power-law model [14–16] , which considers thefluid’s non- Newtonian featuresincluding shear thinning and yield stress char- acteristics [17–22] . In the recent few decades, the investigation of fluid dynamics and heat transfer in the non-Newtonian fluid flow has been identified as one of the most important issues for re- searchers. In points of the fact, comprehension of the property of non-Newtonian fluid movement and the thermal-mechanical be- havior of the fluid stream can result in a better understanding of scientific phenomena occurring in real life [23–28] , which has trig- gered many investigators in different branches of engineering to concentrate on the non-Newtonian fluid flow simulation with dif- ferent methodologies.

Up to the present, a number of studies on the behavior and characteristics of non-Newtonian fluid flow under different bound- ary conditions have been explored. Rios-Iribe et al. [29] proposed a precise numerical method to examine a non-Newtonian fluid flow’s heat and mass transfer under constant wall temperature condi- tions. The non-Newtonian power-law fluid model was used to eval- uate the shear-thinning behavior of the fluid. They comprehen- sively evaluate the respective distributions of the flow field, the local heat transfer coefficient, and the Nusselt number, considering the effects of crucial parameters, including the Reynolds, Prandtl, and Fanning friction numbers. Nadeem et al. [17] analytically stud- ied the blood flowing through a tapered stenosed artery. By assum- ing the steady flow and describing the blood rheology through the non-Newtonian power-law fluid model, they attempted to achieve the exact solution for velocity, resistance impedance, wall shear stress, and shearing stress at the stenos in the throat. Zaman et al. [3] employed an explicit finite difference method to analyze the dynamic response of heat and mass transfer to the bloodstream in- side a tapered artery. The blood was assumed to be a generalized power-law fluid. They quantitatively inspected the blood flow char- acteristics such as velocity distribution, temperature and concen- tration profiles, volumetric flow rate, and arterial wall shear stress for various emerging parameters. Toghraie et al. [30] carried out a numerical approach to simulate the blood flow inside the artery with different radiuses under constant heat flux. They determine velocity and temperature distributions of flow and Nusselt num- ber in axial and radial directions. This study concludes that blood temperature improves with increasing longitudinal and radial dis- tances. In this regard, the flow temperature reaches a maximum value at maximum axial and radial positions from references to the entrance and central zones of the artery. Foong et al. [31] nu- merically investigate the thermal characteristics of blood stream- ing inside the body vessel under prescribed isoflux thermal bound- ary conditions employing Newtonian and non-Newtonian models. The obtained results show that the maximum temperature of non- Newtonian blood flow is higher than that of Newtonian blood flow, meaning that the kind of Newtonian and non-Newtonian fluid ap- proach significantly affect the thermal characteristic of blood inside the artery. Saeed Khan and Ali [32] conducted a semi-analytical approach to assess the Graetz problem for an inelastic power-law fluid through ducts under both isothermal and isoflux boundary conditions in the presence of viscous dissipation. The average tem- perature and local Nusselt number profiles were obtained consider- ing the influences of the critical factors such as slip length, power- law index, and Brinkman number. The results revealed that en- hancing the shear-thinning behavior of the fluid result in an in- crement in the local Nusselt number.

Based on the literature review, studies on the non-Newtonian nature of heat transfer and thermal characteristics of flow in- side tubes under different operational conditions attract the re- searcher’s attention for further examination. Moreover, previous works mainly concentrate on investigating the dynamics and heat transfer of non-Newtonian fluids streaming inside the tube under isothermal or isoflux boundary conditions. However, the analyses of many real-world applications involving the engineering indus- tries illustrate that heat flux applied on the tube surface is not quite constant but varies along the tube length [ 33 , 34 ]. As a result, considering the non-uniform heat flux boundary conditions can significantly enhance the accuracy of the previous results and pro- vide essential information to better understand fundamental con- cepts of the heat transport phenomenon in non-Newtonian fluid flow through the tube.

The current inquiry points to scrutinizing a shear-thinning non- Newtonian fluid streaming through a tube beneath axisymmetric geometry by utilizing mechanical engineering science. In this re- spect, the non-Newtonian rheology of the considered fluid is mod- eled by a power-law pattern. Also, non-uniform heat fluxes that sinusoidally vary along the tube length are imposed on the tube walls to enhance the simulation’s accuracy. A closed-form solution for laminar heat transfer to power-law fluid with different power- law indexes in a circular tube is obtained by employing the sepa- ration of variables (SOV) method. The resulting Eigen functions and Eigen values are obtained using MATLAB built-in solver BVP4C. In the meantime, an approximate integral technique is conducted to portray the non-Newtonian nature of fluid and specify heat trans- fer characteristics of flow. The flow’s axial and radial temperature distributions in both entrance and fully developed zones are deter- mined quantitatively for various values of heat flux applied on the wall boundaries of the tube and power-law index. Subsequently, local and average Nusselt numbers in both entry and fully devel- oped regions are determined. Additionally, the effect of the power- law index on local Nusselt number and the difference between wall and bulk temperatures is examined based on the results ob- tained through both approximate and exact solutions.

Fig. 1 illustrates the structure and the coordinates of the consid- ered system. As can be seen in this figure, a homogenous, incom- pressible, and non-Newtonian power-law fluid of constant phys- ical properties flows inside a smooth tube under applying non- uniform heat flux on the boundary walls. In this study, the cylin- drical polar coordinate (r,θ,z) system is utilized so as to analyze the flow behavior, wherer,θ, andzdenote the radial, circumferen- tial, and longitudinal directions, orderly. The fluid enters atz= 0 with inlet temperatureT0 , which is considered to be 310 K. The tube wall is presumed to be a rigid cylindrical channel. It is also assumed that the flow in the entry is thermally developing but hydro-dynamically developed. The influence of viscous dissipation and heat generation on heat transfer is neglected in the investi- gated system. Other assumptions taken into account in this analy- sis are Steady-state, laminar flow, axisymmetric flow, Parallel flow, constant density, no gravity effects, and high Peclet number (Pe>100).

Fig. 1. Schematic of non-Newtonian fluid flow through tubeconsideringnon- uniform wall heat flux.

It should be noted that blood as a shear-thinning non- Newtonian fluid is employed in this investigation, whose principal characteristics are listed in Table 1 .

Table 1 Thermo-physical properties of blood at 310 K [30] .

In the current work, the power-law model is considered a means to characterize the shear-thinning nature of the considered fluid inside a tube. For a power-law fluid, the shear stress is ex- pressed by [ 35 , 36 ]

where

Substituting Eq. (2) into Eq. (1)

Expecting that velocity declines with an increment of radius so that<0 , set

wherendenotes the flow behavior index. Forn= 1 the power-law model regards the Newtonian model. However, whenn<1 the fluid is called pseudoplastic or shear-thinning, and whenn>1, it is called dilatant or shear-thickening [37] .

The flow field is obtained by solving the conservation of mass and momentum equations [38–40] .The continuity equation in cylindrical coordinates can be written as

For parallel flow

The Navier-Stokes equation in thez-direction is applied in order to determine the axial component of velocityuz[38] .

Sinceuzis dependent onronly, Eq. (8) reduces to

whereτrzcan be found from Eq. (4) . Also, based on the radial component of the Navier-Stokes equation, the longitudinal pressure gradient in the tube is constant. The axial velocity distribution can be achieved by integrating Eq. (9) . Integrating once

Separating variables

The two boundary conditions onuzare

Equations (10) and (12) givesc1 = 0 .

Integrating again

By applying the boundary condition (Eq. (12)) into Eq. (14) , the velocity distribution is determined as

Mean velocity can be given as

Substituting Eq. (16) into Eq. (15) yields

With the velocity profile now determined, we seek to formulate the energy equation. In this context, the following additional sim- plifications are introduced: steady-state, laminar flow, high Peclet number (Pe>100), and no energy generation.

The energy balance equation [38] in cylindrical coordinates is formulated as

The last term in the above equation denotes dissipation viscos- ity.

For large values of the Peclet number (Pe>100), axial con- duction can be ignored in comparison to radial conduction [41] . Substituting the relation obtained for shear stress (Eq. (4)) into Eq. (18) yields Three boundary conditions are needed to solve the energy equation.

They are:

In order to identify the effects of crucial factors on the ther- mal behavior of the non-Newtonian fluid flow inside the tube, the following dimensionless variables are presented as [42]

Also, the Peclet number can be formulated as [43]

The energy equation is given in dimensionless form as

wherePrandEcindicate Prandtl number and Eckert number, re- spectively. They can be defined as follows [ 32 , 44 ]

If it is presumed that the Eckert number multiplied by the Prandtl number (i.e., Brinkman number) is small compared to unity (Br=Pr·Ec1), the viscous dissipation is negligible [45] .Therefore, the energy equation reduces to

subject to the boundary conditions

In order to solve the PDE (Eq. (31)) by using the separation of variables method, all boundary conditions in r direction must be homogeneous [46] . Asz′ → ∞ the flow would be thermally fully de- veloped. As a result, we use superposition of the form

whereθ+(r′,z′)takes the homogenous form of the boundary con- ditions, and whereθFD(r′,z′)refers to the problem that flow is thermally fully developed.

For the purpose of determining the fully developed temperature profileθFD(r′,z′), we introduce a non-dimensional temperature as [47]

Fully developed temperature is described as a profile in which?(r′)does not depend onz′ . Hence,

Newton’s law of cooling is defined as [48–50]

It is noteworthy thatTs(z′)andTm(z′)are unknown. Nonethe- less, sinceq′′sandhare constant, it follows from Eq. (38) that

Substituting Eq. (39) into Eq. (37)

If the variations in kinetic and potential energy are neglected, the application of energy conservation for the element under the steady-state condition gives

In the dimensionless form

wherepis tube perimeter.

According Eqs. (40) and (42) we have

Substituting Eq. (43) into the energy equation gives

The solution can be obtained as follows

Applying the boundary condition, we have

The integration constantg(z′)representing the local centerline temperatureθ(r′ = 0,z′)cannot be obtained by using the bound- ary condition (Eq. (34)) since Eq. (45) already satisfies this con- dition. Accordingly, the mean (bulk) dimensionless temperatureθm(z′)is utilized to determineg(z′).

Substituting Eqs. (17) and (45) into Eq. (47)

On the other hand, from Eq. (42) we know

Thus,

Consequently, the fully developed temperature can be written as follows

With the fully developed temperature distribution now ob- tained, we express the energy transport equation as

with the homogenous boundary conditions inr′ -direction

We now solve the PDE (Eq. (52)) by assuming the separation of the form

Substituting the above into the PDE, separating variables, and setting the separated equation equal to ?λ2, yields

We consider first the ODE depending onz′ :

which yields the expected decaying exponential solution

We now consider ther′ variable of the equation

Equation (60) is of the Sturm-Liouville type, and thus Eigen valuesλmwith associated Eigen functionsR1mare orthogonal re- garding the calculated weight functionThe equa- tion can be solved numerically using the following boundary con- ditions.

The general solution can be obtained by a summation of all so- lutions as follows.

The final step is to determine the constantcmusing the non- homogeneous boundary condition atz′ = 0 given by Eq. (55) , which yields

Orthogonality can be applied to determinecm.

We can now superimpose the two solutions Eqs. (51) and (62) to generate the desired solution:

Surface temperature is determined by settingr′ = 1 in the above relation

It is worth mentioning that eigen valuesλm, eigen functionsR1m, and constantcmcan be numerically determined using MAT- LAB software.

The local Nusselt number based on diameter is [38]

Substituting the dimensionless mean and surface temperatures Eqs. (49) and (66) into the above formula, the local Nusselt number can be obtained as

Also, the average Nusselt number can be determined according to the following formula [38]

Duhamel’s theorem can be employed so as to generalize the re- sults achieved for uniform wall heat flux to the situation that the heat flux is arbitrary. If we consider that an axial heat flux varia- tionq′′s(z′)is applied to the surface of the tube, the temperature distribution of the flow can be obtained as follows.

By assuming that the thermal boundary flux into the blood is specified and varies sinusoidally along the tube lengthLas

q′′0sinand substituting Eq. (65) into the above formula, we

whereGzdenoting Graetz number is defined as

Finally, the temperature profile for the case in which wall heat flux sinusoidally varies along the tube length is determined as fol- lows.

The analysis is now performed in two distinct regions: the first region, during which the two thermal boundary layers do not merge (i.e.,Δ′(z′)≤1), and the second region, during which the longitudinal distance exceeds the location at whichΔ′(z′)= 1 .

Note thatΔ′=is thermal boundary layer in non-dimensional form.

?The first region: entrance region

Integrating Eq. (31) with respect tor′ from 1 ?Δ′(e.g., edge of the thermal boundary layer) to 1 yields

We assume a second-degree polynomial for the temperature profile over the thermal layer as

which satisfies the following known exact and approximate bound- ary conditions

Substituting the temperature profile into the energy integral equation gives

The above ODE can be evaluated numerically noting thatΔ′(z′ = 0)= 0 .

OnceΔ′(z′)is determined, the temperature distribution for the boundary layer can be obtained as a function ofr′ andz′ .This so- lution is valid for 1 ?Δ′≤r′ ≤1 , as long asΔ′(z′)≤1.

Note that the temperature profile corresponding to the first re- gion is not applicable for distancesz′ ≥z′L.The entrance lengthz′Lis the location where two boundary layers merge. The laminar en- trance length can be predicted by settingΔ′(z′ =z′L)= 1 .

?The second region: Fully developed region

For distancesz′ ≥z′L, the concept of the thermal layer has no physical significance. The analysis for the second region may be performed in the following manner.

Integrating Eq. (31) on the interval 0 ≤r′ ≤1 , gives

we choose a quadratic polynomial representation in the form have:

which satisfies the following boundary conditions

Substituting the temperature profile into the energy integral equation gives

Comparing the respective temperature profiles of first and sec- ond regions (Eqs. (75) and (81)) atz′ =z′Lyields

The first-order ordinary differential equation for theθs(z′)can be solved subjected to the above boundary condition.

Consequently, the temperature distribution in the second region for distancesz′ ≥z′Ldefines as

Furthermore, the local and average Nusselt numbers can be cal- culated based on Eqs. (67) and (69) , respectively.

Figure 2 a and 2 b depict the local and average Nusselt numbers as a function of axial distance for various values of power-law in- dexn, respectively. As can be pointed out from these plots, the local Nusselt number drastically declines by marching from the in- let until reaching its corresponding asymptotic value far down the tube. As expected, both local and average Nusselt numbers exhibit a similar trend in the considered interval of axial distances. How- ever, the average Nusselt number is higher than the local Nusselt number at specific values of axial distance. It can be seen from these figures that in the entry region, the maximum Nusselt num- ber corresponds to then= 0.5. As the power-law index of the fluid gets smaller/bigger than this critical value (i.e.,n= 0.5), the Nusselt number reduces from its maximum magnitude. Neverthe- less, decreasing the power-law index from 1 to 0.25 improves the asymptotic Nusselt number when the flow becomes fully devel- oped. In point of fact, these figures confirm the higher fully de- veloped Nusselt number values for shear-thinning blood flow com- pared to the Newtonian fluid flow. It can also be interpreted from these plots that both Nusselt numbers for the shear-thinning case reach fully developed conditions earlier than the counterparts for the Newtonian case. According to Fig. 2 a, by marching through the tubefromz′ = 0.01 toz′ = 1 , the value of the local Nusselt number varies from 9.04 to 5.31 and from 9.23 to 4.36 in the case that the power-law index is 0.25 and 1, respectively.

Dimensionless temperature as a function of radial distance is demonstrated in Fig. 3 for several longitudinal distances taking into account power-law index influences. As observable in this fig- ure, with the increasing of radial distance at the certain values of power-law indexnand axial distance, the flow temperature con- tinuously enhances until it reaches the maximum amount at the tube surface. As a matter of fact, the prescribed heat flux at the tube wall augments the temperature of the blood flowing inside the vessel, which shall be a substantial phenomenon in blood oxy- genation. Moreover, increasing the longitudinal distance results in a significant enhancement in the magnitude of flow temperature. Regarding Fig. 3 , at the low values ofr′ (i.e., near the center of the blood vessel), the local temperature of flow is inversely propor- tional to the power-law indexnso that increasing the power-law index would cause a diminution in the temperature of flow. Nev- ertheless, as we move from the centerline toward the tube ^ primes wall, this proportion becomes direct so that promoting the power- law index leads the flow temperature to increase. Furthermore, it is clear that the power-law index has a more impressive effect on the flow temperature at the wall vicinity of the tube than near the center of that. This phenomenon would be important in medical diagnostics of blood disorders. For the power-law indexes of 0.25 and 1, the dimensionless central temperature of the flow is found to be almost 0.93 and 0.91, respectively, whenz′ = 0.5 .

Fig. 2. (a) Variations of local Nusselt number with axial distance considering power-law index effect. (b) Variations of average Nusselt number with axial dis- tance considering power-law index effect.

Fig. 3. Radial temperature variation for several longitudinal distancesconsidering power-law index impact.

Fig. 4. (a) Temperature distribution along the tube wall under both constant and sinusoidally varying wall heat fluxes, n = 0.25. (b) Temperature distribution along the tube wall under both constant and sinusoidally varying wall heat fluxes, n = 0.5. (c) Temperature distribution along the tube wall under both constant and sinusoidally varying wall heat fluxes, n = 0.75. (d) Temperature distribution along the tube wall under both constant and sinusoidally varying wall heat fluxes, n = 1.

Figure 4 a- 4 d indicate the variation of tube wall temperature along the axial length in the presence of uniform and non-uniform heat fluxes for different values of power-law indexes. As presented in these Figures, for the case that constant wall heat flux is sub- jected, the temperature of the tube surface gradually rises un- til approaching its respective maximum magnitude at the end of the tube, which leads to an increase in the blood temperature along the tube length due to the thermal energy transfer. How- ever, switching from constant heat flux to sinusoidally varying heat flux alters remarkably the trend of the wall temperature profile. In this respect, the prescribed thermal boundary flux type is the principal control parameter that specifies the severity and trend of wall temperature variation. Also, in the case that the wall heat flux varies sinusoidally along the tube length, a diminution in the am- plitude of sinusoidal function heat flux applied on the tube wall strikingly lessens the wall temperature. It can be concluded that the role of the amplitude is notable, especially in the axial dis- tances in which the wall temperature is maximum. As can be seen from these figures, at certain axial distance and heat flux values, the wall temperature for non-Newtonian blood stream is less than the counterpart for the Newtonian case due to differences in their viscous behaviors and reactions in the exposure of tube wall ef- fects. Moreover, in the non-Newtonian situation, a decrement in the power-law index reduces the wall temperature for uniform and non-uniform heat fluxes. With respect to Fig. 4 a, for the case in which the tube wall is put under sinusoidal heat flux with am- plitude of 150 W/m2and 200 W/m2, respectively, the maximum temperature of the tube wall is 311.17 K and 311.56 K. It can be inferred that the investigated range of heat fluxes can increase the blood temperature as much as 1.56 K, which is dangerous for hu- man health.

Fig. 5. Local Nusselt number variations against axial distance, obtained by both exact and approximate solutions.

The change of Local Nusselt number along the tube length, achieved by the exact and approximate solutions, is presented in Fig. 5. This figure confirms that the results obtained through the approximate integral method are closely compatible with those ob- tained by the exact solution, which adequately substantiates ver- ification of the applicability of the approximate integral method. It can be implied from this plot that the maximum errors of the approximate analytic results (compared to the exact method re- sults) relate to the Newtonian case whose exact solution is ob- tained by Ref. [51] . For the non-Newtonian blood flow, the accuracy of predicted values of Nusselt numbers improves by decreasing the power-law indexes.

The limiting value of the Nusselt number calculated by both ex- act and approximate solutions for different quantities of power-law indexes is listed in Table 2 . Based on the comparison, suitable com- patibility is observed between the results of approximate and ex- act solutions with an average difference of 7.4%. With respect to this table, for the shear-thinning non-Newtonian fluid (0

Table 2 Comparison of exact and approximate results of asymptotic Nusselt number.

Figure 6 delineates the variation of the difference between wall temperature and bulk temperature with the axial position for the various magnitude of power-law indexes. Regarding this figure, by going through the tube, the difference between wall temper- ature and bulk temperature constantly enhances until it asymp- totically approaches its limiting value far down the tube. Further- more, this figure shows that the asymptotic difference between wall and bulk temperatures for Newtonian fluid flow is higher than that for shear-thinning blood flow. It can also be pointed out from Fig. 6 that there is a proper agreement between the results achieved by exact and approximate solutions, particularly in the case that blood flow is characterized by the non-Newtonian mod- elwith low power-law indexes. As a point of fact, by reducing the value of the power-law index, the consistency between the results of the two methods significantly improves.

Fig. 6. The difference between wall and bulk temperatures as a function of axial distance, obtained by both exact and approximate solutions.

In the present work, the effort s were devoted to investigating the thermal characteristics and dynamics of the non-Newtonian fluid flow through the tube. In order to accurately simulate the rheological behavior of streaming blood as a shear-thinning non- Newtonian fluid, the power-law paradigm was applied. Boundary walls of the tube are settled under prescribed axially varying heat flux boundary conditions. By assuming a fully developed veloc- ity profile, the development of the temperature profile was an- alyzed, and the flow structure and characteristics of heat trans- fer were elaborately investigated. For this purpose, the momentum and energy conservation equations governing the flow in an ax- isymmetric arrangement are solved by employing both exact and approximate methods. A detailed comparison between the results obtained from the presented two methods under certain condi- tions was conducted. Eventually, the influences of applied heat flux transferred from the tube wall to the blood flow on the axial tem- perature profile were evaluated. The temperature distribution of non-Newtonian blood flow inside the tube with different power- law indexes was studied and compared with that of Newtonian fluid flow. Furthermore, the variation of local and average Nusselt numbers regarding axial distance in both entry and fully developed regions was scrutinized considering the power-law index effects. The results of this study demonstrate the application of mechan- ical engineering in supporting medical science. The following im- portant results are summarized:

?It is found that the fully developed Nusselt number for shear- thinning blood flow has higher values in comparison to the Newtonian fluid flow. According to the results, the value of the asymptotic Nusselt number equals 5.31 and 4.36 in the case that the power-law index is 0.25 and 1, respectively.

?Results show that the maximum temperature value belongs to

the farthest axial and radial distances from the entrance and central zone. For the power-law indexes of 0.25 and 1, the max- imum dimensionless temperature of the flow is found to be around 1.58 and 1.66, respectively.

?Power-law index affects both axial and radial temperature of flow. Based on the results, a decrement in the power-law in- dex reduces the wall temperature for both uniform and non- uniform heat flux cases.

?The findings reveal that the power-law index has a more im- pressive impact on the flow temperature at the wall vicinity of the tube than in the central region of that. This phenomenon would be important in medical diagnostics of blood disorders.

?Switching from constant heat flux to sinusoidally varying heat flux can change the thermal behavior of the blood flow.

?The amplitude of sinusoidal heat flux applied on the tube wall plays a significant role in the axial temperature of the blood stream. The influence of the amplitude becomes more visible in the axial distances where the wall temperature is maximum. With regard to the calculations, for the case that the tube wall is put under sinusoidal heat flux with the investigated range of the amplitude andn= 0.25, the blood temperature can en- hance up to 311.56 K, which is undoubtedly dangerous for hu- man health.

?The results of the approximate integral method are in a sat- isfactory consistency with those of the exact solution, particu- larly for the non-Newtonian blood model with low power-law indexes. Comparing the results for the fully developed Nusselt number obtained by two presented methods gives an average error of 7.4%.However, with decreasing the power-law index, the discrepancy between the two presented methods signifi- cantly reduces.

?Reducing the power-law index lessens the asymptotic differ- ence between wall and bulk temperatures and subsequently in- creases the asymptotic Nusselt number. Fully developed Nusselt number forn= 0.25 is 21.79% higher than the counterparts forn= 1. In the entrance region, however,n= 0.5 is the criti- cal power-law index such that as the power-law index of the fluid becomes smaller/bigger than this critical value, the Nus- selt number reduces from its maximum magnitude.

Declaration of Competing Interest

The authors declare that there is no conflict of interests regard- ing the publication of this paper.

Acknowledgment

Shahin Faghiri, Shahin Akbari, Mohammad Behshad Shafii and, Khashayar Hosseinzadeh want to express their gratitude to the Deputy of Research and Technology of Sharif University of Technol- ogy and Sharif Energy, Water and Environment Institute (SEWEI) for providing a suitable working environment to carry out the ex- periments.